II. the balls U and V will fly off towards M and N; and LECT. will raise the weights in the towers at the same instant. This shews, that when bodies of equal quantities of matter revolve in equal circles with equal velocities, their centrifugal forces are equal. 5. Take away these equal balls, and instead of them, put a ball of six ounces into the bearer M X, at a sixth part of the distance w z from the center, and put a ball of one ounce into the opposite bearer, at the whole distance ry, which is equal to w z from the center of the bearer; and fix the balls at these distances on their cords, by the screw nuts at top; and then the ball U, which is six times as heavy as the ball V, will be at only a sixth part of the distance from its center of motion; and consequently will revolve in a circle of only a sixth part of the circumference of the circle in which V revolves. Now, let any equal weights be put into the towers, and the machine be turned by the winch; which (as the catgut string is on equal wheels below) will cause the balls to revolve in equal times; but V will move six times as fast as U, because it revolves in a circle of six times its radius; and both the weights in the towers will rise at once. This shews, that the centrifugal forces of revolving bodies (or their tendencies to fly off from the circles they describe) are in direct proportion to their quantities of matter multiplied into their respective velocities; or into their distances from the centers of their respective circles. For, supposing U, which weighs six ounces, to be two inches from its center of motion w, the weight multiplied by the distance is 12; and supposing V, which weighs only one ounce, to be 12 inches distant from the center of motion, the weight 1 ounce multiplied by the distance 12inches is 12. And as they revolve in equal times, their . velocities are as their distances from the center, namely, as 1 to 6. LECT. II. A double velocity in the same balance to If these two balls be fixed at equal distances from their respective centers of motion, they will move with equal velocities; and if the tower O has 6 times as much weight put into it as the tower P has, the balls will raise their weight exactly at the same moment. This shews that the ball U, being six times as heavy as the ball V, has six times as much centrifugal force, in describing an equal circle with an equal velocity. 6. If bodies of equal weights revolve in equal circles with unequal velocities, their centrifugal forces are as the circle, is a squares of the velocities. To prove this law by an expea quadru- riment, let two balls U and V of equal weights be fixed. ple power on their cords at equal distances from their respective of gravity. centers of motion w and x; and then let the catgut string E be put round the wheel K (whose circumference is only one half of the circumference of the wheel H or G) and over the pulley s to keep it tight; and let four times as much weight be put into the tower P, as in the tower O. Then turn the winch B, and the ball V will revolve twice as fast as the ball U in a circle of the same diameter, because they are equidistant from the centers of the circles in which they revolve; and the weights in the towers will both rise at the same instant, which shews that in a double velocity the same circle will exactly balance a quadruple power of attraction in the center of the circle. For the weights in the towers may be considered as the attractive forces in the centers, acting upon the revolving ball; which, moving in equal circles, is the same thing as if they both moved in one and the same circle. Kepler's 7. If bodies of equal weights revolve in unequal circles, in such a manner that the squares of the times of their going round are as the cubes of their distances from the centers of the circles they describe; their centrifugal forces are inversely as the squares of their distances from those centers. For, the catgut string remaining 1 as in the last experiment, let the distance of the ball V LECT. from the center x be made equal to two of the cross divisions on its bearer; and the distance of the ball U from the center w be three and a sixth part; the balls themselves being of equal weights, and V making two revolutions by turning the winch, in the time that U makes one so that if we suppose the ball V to revolve in one second, the ball U will revolve in two seconds, the squares of which are one and four; for the square of one is only one, and the square of two is four; therefore the square of the period or revolution of the ball V, is contained four times in the square of the period of the ball U. But the distance of V is 2, the cube of which is 8, and the distance of U is 3, the cube of which is 32 very nearly, in which 8 is contained four times; and therefore, the squares of the periods of V and U are to one another as the cubes of their distances from x and w, which are the centers of their respective circles. And if the weight in the tower O be four ounces, equal to the square of 2, the distance of V from the center x; and the weight in the tower P be 10 ounces, nearly equal to the square of 3 the distance of U from w; it will be found upon turning the machine by the winch, that the balls U and V will raise their respective weights at the same instant of time. Which confirms that famous proposition of KEPLER, viz. That the squares of the periodical times of the planets round the sun are in proportion to the cubes of their distances from him; and that the sun's attraction is inversely as the square of the distance from his center that is, at twice the distance, his attraction is four times less; and thrice the distance, nine times less; at four times the distance, sixteen times less; and so on, to the remotest part of the system. LECT. II. 8. Take off the catgut string E from the great wheel D and the small wheel H, and let the string F remain Take away also the bearer The absur- upon the wheels D and G. vortexes. chine AB upon it, fixing this a Б such a manner, that the end ef may rise above the board to an angle of 30 or 40 degrees." In the upper side of this machine are two glass tubes a and b, close stopped at both ends; and each tube is about three quarters full of water. In the tube a is a little quicksilver, which naturally falls down to the end a in the water, because it is heavier than its bulk of water; and on the tube b is a small cork which floats on the top of the water at e, because it is lighter; and it is small enough to have liberty to rise or fall in the tube. While the board b, with this machine upon it, continues at rest, the quicksilver lies at the bottom of the tube a, and the cork floats on the water near the top of the tube b. But, upon turning the winch, and putting the machine in motion, the contents of each tube will fly off towards the uppermost ends (which are farthest from the center of motion) the heaviest with the greatest force. Therefore the quicksilver in the tube a will fly off quite to the end f, and occupy its bulk of space there, excluding the water from that place, because it is lighter than quicksilver; but the water in the tube b flying off to its higher end e, will exclude the cork from that place, and cause the cork to descend towards the lowermost Note 24. A better mode of performing this experiment consists in screwing a hollow globe to the whirling table. If this be half filled with water, and a wax taper placed upon a cork float in the center, the water, on being whirled, will rise above its previous level, and occupy the equator of the globe, so that the taper will be seen to burn beneath the water, which will thus form a fluid wall around it several inches in height. end of the tube, where it will remain upon the lowest end of the water near b; for the heavier body, having the greater centrifugal force, will therefore possess the uppermost part of the tube; and the lighter body will keep between the heavier and the lowermost part. This demonstrates the absurdity of the Cartesian doctrine of the planets moving round the sun in vortexes; for, if the planet be more dense or heavy than its bulk of the vortex, it will fly off therein, farther and farther from the sun; if less dense, it will come down to the lowest part of the vortex, at the sun and the whole vortex itself must be surrounded with something like a great wall, otherwise it would fly quite off, planets and all together. But while gravity. exists, there is no occasion for such vortexes; and when it ceases to exist, a stone thrown upwards will never return to the earth again. LECT. II. 9. If a body be so placed on the whirling board of If one bothe machine, that the center of gravity of the body dy moves 25 round ano of them must move common be directly over the center of the board, and the board ther, both be put into ever so rapid a motion by the winch B, the of t body will turn round with the board, but will not re- round their move from the middle of it; for, as all parts of the body center of are in equilibrio round its center of gravity, and the gravity. center of gravity is at rest in the center of motion, the centrifugal force of all parts of the body will be equal at equal distances from its center of motion, and therefore the body will remain in its place. But if the center of gravity be placed ever so little out of the center of motion, and the machine be turned swiftly round, the body will fly off towards that side of the board on which its center of gravity lies. Thus, if the wire C with its little ball B be taken away from the demiglobe A, and the flat side ef of this demi-globe be laid upon B C Note 25. See engraving, page 33, A |